We reelaborate on the basic properties of lossless multilayers. We show that the transfer matrices for these multilayers have essentially the same algebraic properties as the Lorentz group SO(2, 1) in a (2 + 1)-dimensional space-time as well as the group SL(2, R) underlying the structure of the ABCD law in geometrical optics. By resorting to the Iwasawa decomposition, we represent the action of any multilayer as the product of three matrices of simple interpretation. This group-theoretical structure allows us to introduce bilinear transformations in the complex plane. The concept of multilayer transfer function naturally emerges, and its corresponding properties in the unit disk are studied. We show that the Iwasawa decomposition is reflected at this geometrical level in three simple actions that can be considered the basic pieces for a deeper understanding of the multilayer behavior. We use the method to analyze in detail a simple practical example.

We elaborate on the consequences of the factorization of the transfer matrix of any lossless multilayer in terms of three basic matrices of simple interpretation. By considering the bilinear transformation that this transfer matrix induces in the complex plane, we introduce the concept of multilayer transfer function and study its properties in the unit disk. In this geometrical setting, our factorization translates into three actions that can be viewed as the basic components for understanding the multilayer behavior. Additionally, we introduce a simple trace criterion that allows us to classify multilayers into three types with properties closely related to one (and only one) of these three basic matrices. We apply this approach to analyze some practical examples that are typical of these types of matrices.

Periodic layered media can reflect strongly for all incident angles and polarizations in a given frequency range. Quarter-wave stacks at normal incidence are commonplace in the design of such omnidirectional reflectors. We discuss alternative design criteria to optimize these systems.

The action of any lossless multilayer is described by a transfer matrix that can be factorized in terms of three basic matrices. We introduce a simple trace criterion that classifies multilayers in three classes with properties closely related with one (and only one) of these three basic matrices. PACS numbers:Any linear system with two input and two output channels can be described in terms of a 2 × 2 transfer matrix [1,2]. In many instances (e.g., in polarization optics [3]), we are interested in the transformation properties of quotients of variables rather than on the variables themselves. Then, the transfer matrix induces a bilinear transformation that represents the action of the system in the complex plane [4].For layered media the use of 2 × 2 matrix methods is standard, but the corresponding complex-plane representation is not a common tool. In fact, it has been recently established that for a lossless multilayer the transfer matrix is an element of the group SU(1,1) [5], which is the key for a deeper understanding of its behavior and simplifies exceedingly its description in terms of bilinear transformations.Moreover, as we have pointed out recently [6], the Iwasawa decomposition provides a remarkable factorization of the transfer matrix representing any multilayer (no matter how complicated it could be) as the product of three matrices of simple interpretation. At the geometrical level, such a decomposition translates directly into three actions that are the basic bricks from which any multilayer action is built.In this Letter we go one step further and show that the trace of the transfer matrix allows for a classification of multilayers in three disjoint classes with properties very close to those appearing in the Iwasawa decomposition.To maintain the discussion as self contained as possible we briefly summarize the essential ingredients of multilayer optics we shall need for our purposes. The configuration we consider is a stratified structure that consists of a stack of plane-parallel layers sandwiched between two semi-infinite ambient (a) and substrate (s) media, which we shall assume to be identical, since this is the common experimental case. Hereafter all the media are supposed to be lossless, homogeneous, and isotropic.We assume an incident monochromatic, linearly polarized plane wave from the ambient, which makes an angle θ 0 with the normal to the first interface and has amplitude E (+) a . The electric field is either in the plane of incidence (p polarization) or perpendicular to the plane of incidence (s polarization). We consider as well another plane wave of the same frequency and polarization, and with amplitude E (−) s , incident from the substrate at the same angle θ 0 [7].As a result of multiple reflections in all the interfaces, we have a backward-traveling plane wave in the ambient, denoted E (−) a , and a forward-traveling plane wave in the substrate, denoted E (+) s . It is useful to treat the field amplitudes as the vectorwhich applies to both ambient and substrate media. Then, ...

Abstract:We determine the optimal thicknesses for which omnidirectional reflection from generalized Fibonacci quasicrystals occurs. By capitalizing on the idea of wavelength-and angle-averaged reflectance, we assess in a consistent way the performance of the different systems. Our results indicate that some of these aperiodic arrangements can largely overperform the conventional photonic crystals as omnidirectional reflection is concerned. 143-215 (1996). 6. B. A. van Tiggelen, "Transverse diffusion of light in Faraday-active media," Phys. Rev. Lett. 75, 422-424 (1995). 7. W. Steurer and D. Sutter-Widmer, "Photonic and phononic quasicrystals," J. Phys. D 40, R229-R247 (2007). 8. A. Poddubny and E. Ivchenko, "Photonic quasicrystalline and aperiodic structures," Physica E 42, 1871-1895 (2010). 9. L. Dal Negro and S. V. Boriskina, "Deterministic aperiodic nanostructures for photonics and plasmonics applications," Laser Photon. Rev. 6, 178-218 (2012). 10. P. J. Steinhardt and S. Ostlund, The Physics of Quasicrystals (World Scientific, 1987). 11. M. Senechal, Quasicrystals and Geometry (Cambridge University, 1995). 12. C. Janot, Quasicrystals: A Primer (Oxford University, 2012). 13. Z. V. Vardeny, A. Nahata, and A. Agrawal, "Optics of photonic quasicrystals," Nat. Photonics 7, 177-187 (2013). 14. E. Maciá, "The role of aperiodic order in science and technology," Rep. Prog. Phys. 69, 397-441 (2006 1768-1770 (1985). 17. M. Kohmoto, L. P. Kadanoff, and C. Tang, "Localization problem in one dimension: Mapping and escape," Phys.Rev. Lett. 50, 1870-1872 (1983). 18. M. Kohmoto, B. Sutherland, and K. Iguchi, "Localization in optics: Quasiperiodic media," Phys. Rev. Lett. 58, 2436-2438 (1987). 19. N.-H. Liu, "Propagation of light waves in Thue-Morse dielectric multilayers," Phys. Rev. B 55, 3543-3547 (1997). 20. S. Tamura and F. Nori, "Transmission and frequency spectra of acoustic phonons in Thue-Morse superlattices,"Phys. Rev. B 40, 9790-9801 (1989 1947-1952 (1998). 24. E. Cojocaru, "Forbidden gaps in finite periodic and quasi-periodic Cantor-like dielectric multilayers at normal incidence," Appl. one-dimensional magnonic quasicrystals," J.